How does this simple picture change in a disordered system? For
concreteness, let's consider a simple ``random bond'' Ising model,

where the exchange constants satisfy

with J>0 and random. I'll discuss short-range
correlated disorder, where the are independent random
variables of different links, taken, without loss of generality to
have zero mean. Let's also assume that these random variables are
narrowly (e.g. exponentially or bounded) distributed, with a
variance

The last requirement guarantees the stability of the ordered phase,
since flipping any spin still costs a positive energy because it
breaks a number of bonds, essentially all of which are ferromagnetic.
In the case of an unbounded but narrow distribution, there will be
exponentially rare regions in which spins are flipped in the ground
state, reducing the average magnetization and likewise reducing ,
but not destabilizing the ordered phase. The critical behavior at the
transition can, however, be modified.

To understand coarsening, we need to consider the behavior of a domain
wall in such a random magnet. We will see that it has a much more
dramatic effect than on the true equilibrium properties. The basic
physical reason for this is clear. In the random system, there are
preferred positions for the domain wall, in which the bonds that are
broken by the wall are chosen to lie on ``weak'' links.

To study this, consider the following toy model. Take a large cube of
size , forcing the spins to point up at x=0 and down at x=L.
At T=0, the system must put in a domain wall to accomodate the
anti-periodic boundary conditions. This is the simplest example of a
topological defect, which is defined as a configuration of an
order parameter in which the states at infinity are non-trivially
``twisted''. In fact, such topological defects often lead to elastic
models of the kind we will discuss here. Coming back to the
interface, there are an enormous choice of possible conformations for
the wall to take. The system must therefore ``solve'' an extremely
non-trivial optimization problem to decide where it should go. I
would like to describe this as a balance between the elastic energy,
which tries to minimize its area (i.e. keep the total number of broken
bonds fixed), and the disorder, which tries to distort the wall to
take advantage of spatially separate weak links.

It is easy to see that the dynamics of such a domain wall will
also be dramatically modified. Because an optimized wall has found at
least a local energy minima, an applied force from, for instance,
curvature forces or an applied field, will have to overcome the
restoring forces near this minima in order for the wall to move.
There are therefore substantial barriers to the wall's movement, and we
expect the resulting motion to be reduced. In fact, we will see that
this slowing down of the coarsening process is incredibly drastic!